Continuum mechanics and normal shear stress

[Niles]

Homework Statement

I am self-studying this note and I am stuck in the derivation of the normal shear stress. I can't see how the relations (23) and (24) come about, i.e. I don't understand

$$\tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx}$$
and
$$\tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}$$

Can someone elaborate on the note to make it clearer? Thanks in advance.

 

[Chestermiller]

Homework Statement

I am self-studying this note and I am stuck in the derivation of the normal shear stress. I can't see how the relations (23) and (24) come about, i.e. I don't understand

$$\tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx}$$
and
$$\tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}$$

Can someone elaborate on the note to make it clearer? Thanks in advance.

Let me guess. The primed stresses are the stresses for a set of axes oriented at 45 degrees to the x and y axes. Correct?

 

[Niles]

Let me guess. The primed stresses are the stresses for a set of axes oriented at 45 degrees to the x and y axes. Correct?

YES! How did you know that? And are we allowed to "add" stresses like this?

 

[Chestermiller]

YES! How did you know that? And are we allowed to "add" stresses like this?

This is just a transformation of the stress tensor components from one set of coordinate axes to another set of coordinate axes. It is analogous to the transformation of a vector in component form from one set of coordinates (i.e., using one set of unit vectors) to another set of coordinates (using another set of basis vectors). Do you know the general transformation relationship for transforming the coordinates of the stress tensor from one set of cartesian coordinates to another set of cartesian coordinates which are rotated through an angle θ relative to the first set?

Chet